ABSTRACT
In Chapters 5-7 chaotic behaviors have been described, mainly associated with nonlinear carrier transport via the impact-ionization avalanche in various bulk-semiconductor materials. Sections 8.1 and 8.2 of this final chapter, will introduce up-to-date problems related to nonlinear carrier transport and chaos, focusing especially on the spatial and spatiotemporal behaviors of current filaments. In section 8.3, chaos in device structures is briefly covered. Among a huge number of nonlinear dynamical systems in biology, chem-
istry, physics and engineering, many systems exhibit universal behaviors of bifurcation and chaos when control parameters are adequately and precisely varied. Such a universality is quite surprising because each dynamical system is almost uncorrelated with others except for the nonlinearity character. In spatially extended systems, it is well recognized that pattern formations and spatiotemporal behaviors are highly complex and in many cases different from each other. The precise differences among many dynamical systems may be termed individuality. The target pattern in the BelousovZhabotinsky (BZ) reaction system and the roll pattern in Rayleigh-Be´rnard convection are different in their physical origin. In fluid dynamical systems, pattern formations are quite different, depending on the types of convection instability and their boundary conditions. Fractals in nature provide more examples, which are various growth patterns (trees, rivers, snow crystals, dendrite patterns of bacteria). However, it is also true that similarity in pattern formations can be seen among different physical systems. Roll patterns in liquid crystal (Williams domain) [1] and Rayleigh-Be´rnard convection [2] are quite similar, where the rich pattern structures in the
liquid crystal can be essentially described by electrohydrodynamic instability, the fundamental equations for which are highly complex. The similarity may be related to a strong analogy between the two systems; the motional force (buoyancy), the viscosity of the fluid, the bifurcation parameter (temperature difference or Rayleigh number) and the Prandtl number in the Rayleigh-Be´rnard convection correspond to the electrostatic force, the elasticity, the voltage ðV2Þ and the driving frequency (by ð f=fcÞ1Þ in the liquid crystal. The spiral patterns in the BZ reaction system resemble the wave pattern of chemotactic activity in dense cell layers of slime mold. Such a variety of pattern formations and spatiotemporal behaviors is not so easy to understand. Although the universal rule of low-dimensional chaos has observed among many dynamical systems and the chaos theory has been now fully established and has prevailed, these systems also exhibit many aspects of unresolved spatiotemporal complexities. The goal may be the full understanding of spatiotemporal complex behaviors. Historically, extensive studies on spatiotemporal behaviors (pattern dynamics) have been done in the fluid system [3], the BZ system [4] and liquid crystal [1]. In semiconductors, the most prominent examples of spatial patterns are current filaments and high-field domains. Stable filaments and domains have been well known for several decades from experimental and theoretical investigations [5, 6]. However, only recently has attention been paid to the field of pattern dynamics, mainly due to the difficulty in visualizing time-resolved spatial patterns. Many more problems remain unanswered than resolved.
